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The $d_2$-transformation for infinite double series and the $D_2$-transformation for infinite double integrals

Authors: Chen Greif and David Levin
Journal: Math. Comp. 67 (1998), 695-714
MSC (1991): Primary 65B10; Secondary 40B05, 65D30
MathSciNet review: 1464144
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Abstract: New transformations for accelerating the convergence of infinite double series and infinite double integrals are presented. These transformations are generalizations of the univariate $d$- and $D$-transformations. The $D_2$-transformation for infinite double integrals is efficient if the integrand satisfies a p.d.e. of a certain type. Similarly, the $d_2$-transformation for double series works well for series whose terms satisfy a difference equation of a certain type. In both cases, the application of the transformation does not require an explicit knowledge of the differential or the difference equation. Asymptotic expansions for the remainders in the infinite double integrals and series are derived, and nonlinear transformations based upon these expansions are presented. Finally, numerical examples which demonstrate the efficiency of these transformations are given.

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Additional Information

Chen Greif
Affiliation: School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel
Address at time of publication: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T-1Z2
MR Author ID: 622883

David Levin
Affiliation: School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

Received by editor(s): November 21, 1995
Received by editor(s) in revised form: July 19, 1996, and January 8, 1997
Article copyright: © Copyright 1998 American Mathematical Society